A hot air balloon of total mass M (including passengers and luggage) is moving with a downward acceleration of magnitude a. As it approaches a mountain, the captain needs to accelerate upwards. He decides to throw enough ballast over board to achieve an upward acceleration of magnitude a/2. What fraction of the initial mass does he have to drop?

When balloon moves in the downward direction two forces acts on it.  

    i)Force exerted by air in the upward direction  

    ii) Weight  

According to newton’s second law of motion:  

      Sum of forces = Ma  

                W – F = Ma  

               Mg – F = Ma     …….. (i)  

when some of the mass m is dropped and balloon is moving in upward direction with acceleration a/2 then,  

               F – W = (M-m)a/2

               F – (M-m)g = (M-m)a/2

               F – Mg + mg = Ma/2 – ma/2 ….. (ii)  

Adding equation (i) and (ii)  

              mg = M(3a/2) – ma/2

              m(g + a/2) = M(3a/2)

              m = M(3a/2)/(g + a/2)

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